Download WORKING SEMINAR ON THE STRUCTURE OF LOCALLY

WORKING SEMINAR ON THE STRUCTURE OF LOCALLY COMPACT
GROUPS
Draft by T. Dumont, Spring 2014, EPFL.
2. Hilbert’s fifth problem and its solutions
Thibaut Dumont, the 3rd of April 2014.
“What is the physicist’s definition of a group?
A Lie group without the manifold structure.”
— An anonymous mathoverflow.net user.
In this second lecture, we present Hilbert’s fifth problem, its interpretations and solutions.
This talk was largely inspired by the book of Tao. Also Montgomery and Zippin wrote a nice
book on the topic.
2.1. Hilbert’s fifth problem. In the beginning of the 20th century, Hilbert addressed a list
of important open problems; the fifth one asks to characterize Lie groups among the topological
groups. The problem was stated in a general form which lead to several interpretations of it.
The first results were obtained by von Neumann and Pontryagin for compact groups and abelian
groups respectively. Later, in 1950, Gleason, then Montgomery and Zippin, obtained a complete
solution to one interpretation of Hilbert’s fifth problem. Three years later, Yamabe gave what
is considered as the final answered to the topic.
In this section, we present the problem asked by Hilbert in 1900. Then we expose the solution
of Gleason and Montgomery-Zippin. The second part treats Yamabe’s improvement of these
results and we finally give a list of examples. In a futur presentation we could discuss the work
of von Neumann and Pontryagin, as well as some representation theory. This should close the
topic of Lie groups and we shall focus more on totally disconnected groups afterward.
Definition 1 (Topological structures).
(i) A topological space X is called locally Euclidean it is locally homeomorphic to Rn , i.e.
every point x ∈ X has an open neighbourhood homeomorphic to an open subset of some
space Rn .
(ii) An n-dimensional (topological) manifold is a Hausdorff locally Euclidean topological space
such that all homeomorphisms in definition (i) are with respect to the same Rn .
(iii) A group G together with a topology is called a topological group if the multiplication and
inverse maps are continuous.
(iv) A locally compact group is a Hausdorff topological group which is a locally compact space,
i.e. every point has a neighbourhood basis consisting of compact neighbourhoods.
(v) A locally Euclidean group is a Hausdorff topological group which is a locally Euclidean
space.
We don’t assume manifolds to be second countable as it is common in the literature. Here is
some straightforward observations and well known results :
• A locally Euclidean space is first countable and locally compact.
• A Hausdorff topological group is first countable if and only if it is metrizable. Therefore
locally Euclidean groups are metrizable.
Date: June 23, 2014.
1
2
WORKING SEMINAR ON THE STRUCTURE OF LOCALLY COMPACT GROUPS
• In a topological group, all translations are homeomorphisms, hence Brouwer’s theorem on
the invariance of the dimension implies that a locally Euclidean group is an n-dimensional
manifold for a fixed n ∈ N. All discrete groups are 0-dimensional manifolds; they are
exactly the 0-dimensional Lie groups.
Definition 2 (Smooth structures). Let M be an n-dimensional manifold.
(i) A smooth atlas is a set of local homeomorphisms (φα )α∈A from open subsets Uα , covering M , into Rn , and such that the transition maps φα ◦ φ−1
β are smooth whenever they
are well defined. Two atlases are compatible if their union is still a smooth atlas. An
equivalence class of atlas for the compatibility relation is called a smooth structure on M .
(ii) An n-dimensional smooth manifold is a n-dimensional manifold together with a smooth
structure. It is clear how to define smooth maps between smooth manifolds and finite
products of manifolds.
(iii) A Lie group is a locally Euclidean group endowed with a smooth structure such that the
multiplication and inverse maps are smooth.
The most commonly accepted interpretation of Hilbert’s fifth problem is to determine whether
the smoothness assumption is redondant in the definition of a Lie group.
Question (Hilbert’s 5th Problem). Are locally Euclidean topological groups in fact Lie groups ?
In other words, is there a smooth structure on a locally Euclidean topological group making
the group operations smooth? The philosophy is that a weak regularity condition such as
continuity and a group-like structure should necessarily come from a stronger regularity, namely
smoothness.
The next proposition and corollary show that if a smooth structure exists it is unique, hence
the problem is well defined. It is a first instance of the aforementioned philosophy.
Proposition 3. Let G, H be Lie groups and Φ : G → H be a continuous homomorphism. Then,
Φ is a smooth map.
Applying this proposition to the identity map of a topological group, we immediately get the
following corollary.
Corollary 4. There is at most one smooth structure on a topological group making it a Lie
group.
Lemma 5 (The exponential map of a Lie group). Let G be a Lie group and denote g its Lie
algebra, i.e. the tangent space at the identity. There is a smooth map exp : g → G satisfying :
(i) exp(sX) exp(tX) = exp((s + t)X), for all s, t ∈ R and X ∈ g,
(ii) exp(X + Y ) = limn (exp(X/n) exp(Y /n))n , for all X, Y ∈ g,
(iii) exp is a local diffeomorphism at 0, i.e. there are U ⊂ g and V ⊂ G open neighbourhoods
of 0 and 1 respectively such that exp |U : U → V is a diffeomorphism.
Definition 6 (One-parameter subgroup). Let G be a topological group. A one-parameter
subgroup φ of G is a continuous group homomorphism φ : R → G.
If G is a Lie group, the exponential map induces an identification between the set of oneparameter subgroups and g given by the formula
φ(t) = exp(tX).
Proof of Proposition 3. Smoothness needs only be verified in a neighbourhood of the identity. The previous identification is now the key. The continuous homomorphism Φ sends
one-parameter subgroups of G to one-parameter subgroups of H. In fact, it defines a linear,
hence smooth, map between the Lie algebras. Composing with respective exponential maps on
sufficiently small neighbourhoods gives the conclusion.
Definition 7 (NSS group). A topological group G has no small subgroup, abbreviated NSS, if
there is a identity neighbourhood U containing no subgroup of G other than {e}.
WORKING SEMINAR ON THE STRUCTURE OF LOCALLY COMPACT GROUPS
3
The study of NSS groups was motivated by the fact that Lie groups are NSS. Once more, the
exponential map plays a crucial role and we see that regularity at the identity is key in these
developments.
Proposition 8. Let G be a Lie groups, then G has NSS.
Proof. Let U ⊂ g be a small neighbourhood of 0 such that the restriction of the exponential
map to it is a diffeomorphism onto its image. We may assume that U is a ball with respect to
some norm on g. The structure of additive group of g implies that given any non-zero vector
X ∈ g, there is an integer n such that nX escapes U , i.e. nX ∈
/ U . Consequently, exp(U ) can
not contain a subgroup of G.
Theorem 9 (Gleason, ∼1950). Let G be a NSS locally Euclidean group. Then G is a Lie group.
This profound result was a major breakthrough and relaunched investigations on Hilbert’s
fifth problem. Indeed, soon afterward Montgomery and Zippin proved that the NSS assumption
was not necessary for locally Euclidean groups, yielding a positive answer to the question asked
by Hilbert nearly fifthy years ago.
Theorem 10 (Montgomery, Zippin, ∼1950). Let G be a locally Euclidean group, then G has
NSS.
Corollary 11 (Hilbert’s fifth problem). Let G be a locally Euclidean group, then G is a Lie
group.
2.2. Approximation by Lie groups. The answer to the first formulation of Hilbert’s fifth
problem was very satisfactory, but one can ask what about more general locally compact group.
How close are they to Lie groups? We present now the results given by Yamabe soon after the
works of Gleason, Montgomery and Zippin. It turns out many locally compact groups can be
approximated by Lie groups. To this purpose we introduce the notion of projective limit.
Definition 12.
(i) Let G = (Gi )i∈I be a family of topological groups indexed on a partially ordered set (I, <)
together with a family of continuous homomorphisms Φ = (ϕij : Gj → Gi ), one for each
pair i < j. We say that (G, Φ) is a projective system if ϕij ◦ ϕjk = ϕik whenever i < j < k.
(ii) The projective limit of a projective system (G, Φ) is defined by
Y
lim Gi := {(gi )i∈I | ϕij (gj ) = gi for all i < j} ⊂
Gi .
←−
i∈I
The product is endowed with the product topology, so that the projective limit with the
subspace topology is a closed subgroup of the product. Therefore if the groups Gi are locally
compact so is lim Gi . Each projection map πj : lim Gi → Gj is a continuous homomorphism
←−
←−
and, for every pair i < j, we have by definition πi = ϕij ◦ πj . The projective limit lim Gi and
←−
the maps (πi )i∈I are universal with respect to the latter property.
If G is the projective limit of a family (Gi )i∈I of Lie groups, it may not be a Lie group.
However, he can be approximated by Lie groups in some sense. The kernel Ki of the projection
πi : G → Gi is a closed normal subgroup of G. Since the map
π i : G/Ki → Gi
is continuous and injective (but not necessarily surjective), a theorem of von Neumann implies
that the quotient G/Ki is a Lie group. Moreover for each pair i < j, we have Kj ⊂ Ki , thus
G/Ki ,→ G/Kj .
“As the kernels Ki shrink to the identity, the Lie groups G/Ki grow to approximate G.”
Theorem 13 (Gleason, Yamabe, 1953). Let G be a locally compact group. Then for every
identity neighbourhood U , there is an open subgroup G0 < G and a compact subgroup K C G0
such that K ⊂ U and G0 /K is a Lie group.
4
WORKING SEMINAR ON THE STRUCTURE OF LOCALLY COMPACT GROUPS
In the connected case, one can always take G0 = G, hence we obtain the following.
Corollary 14 (Connected Case). Let G be a connect locally compact group, then G is a projective
limit of Lie groups,
G = lim G/K,
←−
where the projective system is indexed on the set of compact normal subgroups K of G. Furthermore, if G has NSS then G is a Lie group.
Corollary 15. A locally compact group G has NSS if and only if G is a Lie group.
Proof. A locally compact group containing an open subgroup that is a Lie group is itself a Lie
group.
In the next talk, we will prove van Dantzig’s theorem on totally disconnected locally compact
groups and see how it can be used to improve Theorem 13.
2.3. Examples of Lie groups. The groups Rn , Rn /Zn , GLn (R) and GLn (C), with their usual
topology, are Lie groups. E. Cartan proved that a (topologically) closed subgroup of a Lie group
is a Lie group; adding SLn (C), Un (C), SUn (C), SLn (R), On (R), SOn (R) to the list. Also the
Heisenberg group and one of its quotient, the Heinsenberg-Weil group, are both Lie groups :




1 R R/Z
1 R R
1
R .
1 R ,
H=
H3 = 
1
1
However, H is not a matrix group. In other words, there is no non-trivial continuous injective
homomorphism H → GLn (C), (see p. 9 of Tao’s book).
2.4. Examples of locally compact groups without Lie structure. The topological groups
in the list below are locally compact but not Lie. We shall also see how they fit in GleasonYamabe’s theorem.
(i) The infinite torus G = (R/Z)N with the product topology is connected. Any neighbourhood of the identity contains a subgroup of the form :
n
Y
Y
Gn =
{1} ×
R/Z,
i=1
n<i
for some integer n large enough, i.e. G is not NSS. The quotients G/Gn ∼
= (R/Z)n are the
finite dimensional torus and G is the projective limit of them with respect to the inclusion
maps.
(ii) At the opposite of connected groups, the p-adic integers Zp , the p-adic field Qp and
GLn (Qp ) are non-discrete locally compact totally disconnected groups. The compact open
subgroups pn Zp , for n ∈ N, form a basis of neighbourhood of the identity in Qp and Zp .
These groups are not NSS and they are projective limits of discrete groups :
Qp ∼
Q /pn Zp ,
Zp ∼
Z /pn Zp ∼
Z/pn Z.
= lim
= lim
= lim
←− p
←− p
←−
(iii) The two previous example were genuine projective limits of Lie groups. The last example is
not and the open subgroup G0 in Gleason-Yamabe’s theorem has to be a proper subgroup
of G. Consider the continuous automorphism of the additive group Qp defined by x 7→ px
and form the semi-direct product G = Qp o Z. The underlying topological space is the
product Qp × Z and the group operation is
(x, n)(y, m) = (x + pn y, n + m),
for all x, y ∈ Qp and n, m ∈ Z. The group G is not a Lie group since pn Zp ×{0}, for n ∈ N,
are arbitrary small subgroups. They are all contained in the open identity neighbourhood
U = Zp × {0}. The latter does not contain any non-trivial subgroup that is normal in G.
Indeed, the conjugation by the element (0, 1) acts like the multiplication by p in the Qp
factor. However, G0 = Qp × {0} is an open subgroup which is a projective limit of Lie
groups.